Hanani triple systems on \(v \equiv 1 \pmod 6\) elements are Steiner triple systems having \((v-1)/2\) pairwise disjoint almost parallel classes (sets of pairwise disjoint triples that span \(v-1\) elements), and the remaining triples form a parallel class. Hanani triple systems are one natural analogue of the Kirkman triple systems on \(v \equiv 3 \pmod 6\) elements, which form the solution of the celebrated Kirkman schoolgirl problem. The authors prove that a Hanani triple system exists for all \(v \equiv 1 \pmod 6\) except for \(v \in \{7,13\}\). This result has many applications in design constructions (e.g., for almost resolvable twofold triple systems), and suggests a number of generalizations and open problems, especially on minimum chromatic index of designs.